AISTATS 2021 Papers

We study the problem of corralling stochastic bandit algorithms, that is combining multiple bandit algorithms designed for a stochastic environment, with the goal of devising a corralling algorithm that performs almost as well as the best base algorithm. We give two general algorithms for this setting, which we show benefit from favorable regret guarantees. We show that the regret of the corralling algorithms is no worse than that of the best algorithm containing the arm with the highest reward, and depends on the gap between the highest reward and other rewards.

The Gaussian-smoothed optimal transport (GOT) framework, recently proposed by Goldfeld et al., scales to high dimensions in estimation and provides an alternative to entropy regularization. This paper provides convergence guarantees for estimating the GOT distance under more general settings. For the Gaussian-smoothed $p$-Wasserstein distance in $d$ dimensions, our results require only the existence of a moment greater than $d + 2p$. For the special case of sub-gamma distributions, we quantify the dependence on the dimension $d$ and establish a phase transition with respect to the scale parameter. We also prove convergence for dependent samples, only requiring a condition on the pairwise dependence of the samples measured by the covariance of the feature map of a kernel space. A key step in our analysis is to show that the GOT distance is dominated by a family of kernel maximum mean discrepancy (MMD) distances with a kernel that depends on the cost function as well as the amount of Gaussian smoothing. This insight provides further interpretability for the GOT framework and also introduces a class of kernel MMD distances with desirable properties. The theoretical results are supported by numerical experiments.The Gaussian-smoothed optimal transport (GOT) framework, recently proposed by Goldfeld et al., scales to high dimensions in estimation and provides an alternative to entropy regularization. This paper provides convergence guarantees for estimating the GOT distance under more general settings. For the Gaussian-smoothed $p$-Wasserstein distance in $d$ dimensions, our results require only the existence of a moment greater than $d + 2p$. For the special case of sub-gamma distributions, we quantify the dependence on the dimension $d$ and establish a phase transition with respect to the scale parameter. We also prove convergence for dependent samples, only requiring a condition on the pairwise dependence of the samples measured by the covariance of the feature map of a kernel space. A key step in our analysis is to show that the GOT distance is dominated by a family of kernel maximum mean discrepancy (MMD) distances with a kernel that depends on the cost function as well as the amount of Gaussian smoothing. This insight provides further interpretability for the GOT framework and also introduces a class of kernel MMD distances with desirable properties. The theoretical results are supported by numerical experiments.

We study a family of algorithms, which we refer to as local update methods, generalizing many federated and meta-learning algorithms. We prove that for quadratic models, local update methods are equivalent to first-order optimization on a surrogate loss we exactly characterize. Moreover, fundamental algorithmic choices (such as learning rates) explicitly govern a trade-off between the condition number of the surrogate loss and its alignment with the true loss. We derive novel convergence rates showcasing these trade-offs and highlight their importance in communication-limited settings. Using these insights, we are able to compare local update methods based on their convergence/accuracy trade-off, not just their convergence to critical points of the empirical loss. Our results shed new light on a broad range of phenomena, including the efficacy of server momentum in federated learning and the impact of proximal client updates.

Bilevel optimization problems are receiving increasing attention in machine learning as they provide a natural framework for hyperparameter optimization and meta-learning. A key step to tackle these problems is the efficient computation of the gradient of the upper-level objective (hypergradient). In this work, we study stochastic approximation schemes for the hypergradient, which are important when the lower-level problem is empirical risk minimization on a large dataset. The method that we propose is a stochastic variant of the approximate implicit differentiation approach in (Pedregosa, 2016). We provide bounds for the mean square error of the hypergradient approximation, under the assumption that the lower-level problem is accessible only through a stochastic mapping which is a contraction in expectation. In particular, our main bound is agnostic to the choice of the two stochastic solvers employed by the procedure. We provide numerical experiments to support our theoretical analysis and to show the advantage of using stochastic hypergradients in practice.

Contrastive learning is a family of self-supervised methods where a model is trained to solve a classification task constructed from unlabeled data. It has recently emerged as one of the leading learning paradigms in the absence of labels across many different domains (e.g. brain imaging, text, images). However, theoretical understanding of many aspects of training, both statistical and algorithmic, remain fairly elusive. In this work, we study the setting of time series—more precisely, when we get data from a strong-mixing continuous-time stochastic process. We show that a properly constructed contrastive learning task can be used to the transition kernel for small-to-mid-range intervals in the diffusion case. Moreover, we give sample complexity bounds for solving this task and quantitatively characterize what the value of the contrastive loss implies for distributional closeness of the learned kernel. As a byproduct, we illuminate the appropriate settings for the contrastive distribution, as well as other hyperparameters in this setup.

The continuum-armed bandits problem involves optimizing an unknown objective function given an oracle that evaluates the function at a query point. In the most well-studied case, the objective function is assumed to be Lipschitz continuous and minimax rates of simple and cumulative regrets are known under both noiseless and noisy conditions. In this paper, we investigate continuum-armed bandits under more general smoothness conditions, namely Besov smoothness conditions, on the objective function. In both noiseless and noisy conditions, we derive minimax rates under both simple and cumulative regrets. In particular, our results show that minimax rates over objective functions in a Besov space are identical to minimax rates over objective functions in the smallest Holder space into which the Besov space embeds.

Naively trained neural networks tend to experience catastrophic forgetting in sequential task settings, where data from previous tasks are unavailable. A number of methods, using various model expansion strategies, have been proposed recently as possible solutions. However, determining how much to expand the model is left to the practitioner, and often a constant schedule is chosen for simplicity, regardless of how complex the incoming task is. Instead, we propose a principled Bayesian nonparametric approach based on the Indian Buffet Process (IBP) prior, letting the data determine how much to expand the model complexity. We pair this with a factorization of the neural network’s weight matrices. Such an approach allows us to scale the number of factors of each weight matrix to the complexity of the task, while the IBP prior encourages sparse weight factor selection and factor reuse, promoting positive knowledge transfer between tasks. We demonstrate the effectiveness of our method on a number of continual learning benchmarks and analyze how weight factors are allocated and reused throughout the training.

We study a novel variant of the multi-armed bandit problem, where at each time step, the player observes an independently sampled context that determines the arms’ mean rewards. However, playing an arm blocks it (across all contexts) for a fixed number of future time steps. The above contextual setting captures important scenarios such as recommendation systems or ad placement with diverse users. This problem has been recently studied [Dickerson et al., AAAI 2018] in the full-information setting (i.e., assuming knowledge of the mean context-dependent arm rewards), where competitive ratio bounds have been derived. We focus on the bandit setting, where these means are initially unknown; we propose a UCB-based variant of the full-information algorithm that guarantees a $\mathcal{O}(\log T)$-regret w.r.t. an $\alpha$-optimal strategy in $T$ time steps, matching the $\Omega(\log(T))$ regret lower bound in this setting. Due to the time correlations caused by blocking, existing techniques for upper bounding regret fail. For proving our regret bounds, we introduce the novel concepts of delayed exploitation and opportunistic subsampling and combine them with ideas from combinatorial bandits and non-stationary Markov chains coupling.

Sampling is a popular method for approximate inference when exact inference is impractical. Generally, sampling algorithms do not exploit context-specific independence (CSI) properties of probability distributions. We introduce context-specific likelihood weighting (CS-LW), a new sampling methodology, which besides exploiting the classical conditional independence properties, also exploits CSI properties. Unlike the standard likelihood weighting, CS-LW is based on partial assignments of random variables and requires fewer samples for convergence due to the sampling variance reduction. Furthermore, the speed of generating samples increases. Our novel notion of contextual assignments theoretically justifies CS-LW. We empirically show that CS-LW is competitive with state-of-the-art algorithms for approximate inference in the presence of a significant amount of CSIs.

In this paper we introduce and study the online consistent k-clustering with outliers problem, generalizing the non-outlier version of the problem studied in Lattanzi-Vassilvitskii [18]. We show that a simple local-search based on-line algorithm can give a bicriteria constant approximation for the problem with O(k^2 log^2(nD)) swaps of medians (recourse) in total, where D is the diameter of the metric. When restricted to the problem without outliers, our algorithm is simpler, deterministic and gives better approximation ratio and recourse, compared to that of Lattanzi-Vassilvitskii [18].

We consider off-policy evaluation in the contextual bandit setting for the purpose of obtaining a robust off-policy selection strategy, where the selection strategy is evaluated based on the value of the chosen policy in a set of proposal (target) policies. We propose a new method to compute a lower bound on the value of an arbitrary target policy given some logged data in contextual bandits for a desired coverage. The lower bound is built around the so-called Self-normalized Importance Weighting (SN) estimator. It combines the use of a semi-empirical Efron-Stein tail inequality to control the concentration and Harris’ inequality to control the bias. The new approach is evaluated on a number of synthetic and real datasets and is found to be superior to its main competitors, both in terms of tightness of the confidence intervals and the quality of the policies chosen.

Randomized smoothing is currently the most competitive technique for providing provable robustness guarantees. Since this approach is model-agnostic and inherently scalable we can certify arbitrary classifiers. Despite its success, recent works show that for a small class of i.i.d. distributions, the largest $l_p$ radius that can be certified using randomized smoothing decreases as $O(1/d^{1/2-1/p})$ with dimension $d$ for $p > 2$. We complete the picture and show that similar no-go results hold for the $l_2$ norm for a much more general family of distributions which are continuous and symmetric about the origin. Specifically, we calculate two different upper bounds of the $l_2$ certified radius which have a constant multiplier of order $\Theta(1/d^{1/2})$. Moreover, we extend our results to $l_p (p>2)$ certification with spherical symmetric distributions solidifying the limitations of randomized smoothing. We discuss the implications of our results for how accuracy and robustness are related, and why robust training with noise augmentation can alleviate some of the limitations in practice. We also show that on real-world data the gap between the certified radius and our upper bounds is small.

This papers studies how competition affects machine learning (ML) predictors. As ML becomes more ubiquitous, it is often deployed by companies to compete over customers. For example, digital platforms like Yelp use ML to predict user preference and make recommendations. A service that is more often queried by users, perhaps because it more accurately anticipates user preferences, is also more likely to obtain additional user data (e.g. in the form of a Yelp review). Thus, competing predictors cause feedback loops whereby a predictor’s performance impacts what training data it receives and biases its predictions over time. We introduce a flexible model of competing ML predictors that enables both rapid experimentation and theoretical tractability. We show with empirical and mathematical analysis that competition causes predictors to specialize for specific sub-populations at the cost of worse performance over the general population. We further analyze the impact of predictor specialization on the overall prediction quality experienced by users. We show that having too few or too many competing predictors in a market can hurt the overall prediction quality. Our theory is complemented by experiments on several real datasets using popular learning algorithms, such as neural networks and nearest neighbor methods.

Labeling data for modern machine learning is expensive and time-consuming. Latent variable models can be used to infer labels from weaker, easier-to-acquire sources operating on unlabeled data. Such models can also be trained using labeled data, presenting a key question: should a user invest in few labeled or many unlabeled points? We answer this via a framework centered on model misspecification in method-of-moments latent variable estimation. Our core result is a bias-variance decomposition of the generalization error, which shows that the unlabeled-only approach incurs additional bias under misspecification. We then introduce a correction that provably removes this bias in certain cases. We apply our decomposition framework to three scenarios—well-specified, misspecified, and corrected models—to 1) choose between labeled and unlabeled data and 2) learn from their combination. We observe theoretically and with synthetic experiments that for well-specified models, labeled points are worth a constant factor more than unlabeled points. With misspecification, however, their relative value is higher due to the additional bias but can be reduced with correction. We also apply our approach to study real-world weak supervision techniques for dataset construction.

Decentralized optimization problems frequently appear in the large scale machine learning problems. However, few works work on the difficult nonconvex nonsmooth case. In this paper, we propose a decentralized primal-dual algorithm to solve this type of problem in a decentralized manner and the proposed algorithm can achieve an $\mathcal{O}(1/\epsilon^2)$ iteration complexity to attain an $\epsilon-$solution, which is the well-known lower iteration complexity bound for nonconvex optimization. To our knowledge, it is the first algorithm achieving this rate under a nonconvex, nonsmooth decentralized setting. Furthermore, to reduce communication overhead, we also modifying our algorithm by compressing the vectors exchanged between agents. The iteration complexity of the algorithm with compression is still $\mathcal{O}(1/\epsilon^2)$. Besides, we apply the proposed algorithm to solve nonconvex linear regression problem and train deep learning model, both of which demonstrate the efficiency and efficacy of the proposed algorithm.

Combinatorial bandit models and algorithms are used in many sequential decision-making tasks ranging from item list recommendation to influence maximization. Typical algorithms proposed for combinatorial bandits, including combinatorial UCB (CUCB) and combinatorial Thompson sampling (CTS) do not exploit correlations between base arms during the learning process. Moreover, their regret is usually analyzed under independent base arm outcomes. In this paper, we use Gaussian Processes (GPs) to model correlations between base arms. In particular, we consider a combinatorial bandit model with probabilistically triggered arms, and assume that the expected base arm outcome function is a sample from a GP. We assume that the learner has access to an exact computation oracle, which returns an optimal solution given expected base arm outcomes, and analyze the regret of Combinatorial Gaussian Process Upper Confidence Bound (ComGP-UCB) algorithm for this setting. Under (triggering probability modulated) Lipschitz continuity assumption on the expected reward function, we derive ($O( \sqrt{m T \log T \gamma_{T, \boldsymbol{\mu}}^{PTA}})$) $O(m \sqrt{\frac{T \log T}{p^*}})$ upper bounds for the regret of ComGP-UCB that hold with high probability, where $m$ denotes the number of base arms, $p^*$ denotes the minimum non-zero triggering probability, and $\gamma_{T, \boldsymbol{\mu}}^{PTA}$ denotes the pseudo-information gain. Finally, we show via simulations that when the correlations between base arm outcomes are strong, ComGP-UCB significantly outperforms CUCB and CTS.

We consider a setting where users interact with a collection of N items on an online platform. We are given class labels possibly corrupted by noise, and we seek to recover the true class of each item. We postulate a simple probabilistic model of the interactions between users and items, based on the assumption that users interact with classes in different proportions. We then develop a message-passing algorithm that decodes the noisy class labels efficiently. Under suitable assumptions, our method provably recovers all items’ true classes in the large N limit, even when the interaction graph remains sparse. Empirically, we show that our approach is effective on several practical applications, including predicting the location of businesses, the category of consumer goods, and the language of audio content.

Relationship between agents can be conveniently represented by graphs. When these relationships have different modalities, they are better modelled by multilayer graphs where each layer is associated with one modality. Such graphs arise naturally in many contexts including biological and social networks. Clustering is a fundamental problem in network analysis where the goal is to regroup nodes with similar connectivity profiles. In the past decade, various clustering methods have been extended from the unilayer setting to multilayer graphs in order to incorporate the information provided by each layer. While most existing works assume – rather restrictively - that all layers share the same set of nodes, we propose a new framework that allows for layers to be defined on different sets of nodes. In particular, the nodes not recorded in a layer are treated as missing. Within this paradigm, we investigate several generalizations of well-known clustering methods in the complete setting to the incomplete one and prove consistency results under the Multi-Layer Stochastic Block Model assumption. Our theoretical results are complemented by thorough numerical comparisons between our proposed algorithms on synthetic data, and also on several real datasets, thus highlighting the promising behaviour of our methods in various realistic settings.

Hierarchical clustering is a fundamental task often used to discover meaningful structures in data. Due to the combinatorial number of possible hierarchical clusterings, approximate algorithms are typically used for inference. In contrast to existing methods, we present novel dynamic-programming algorithms for exact inference in hierarchical clustering based on a novel trellis data structure, and we prove that we can exactly compute the partition function, maximum likelihood hierarchy, and marginal probabilities of sub-hierarchies and clusters. Our algorithms scale in time and space proportional to the powerset of N elements, which is super-exponentially more efficient than explicitly considering each of the (2N − 3)!! possible hierarchies. Also, for larger datasets where our exact algorithms become infeasible, we introduce an approximate algorithm based on a sparse trellis that out- performs greedy and beam search baselines.